Law of Total Expectaiton

The expectation of a distribution can be seen as the weighted sum

Theorem

$$E(Y)=E[E(Y|X)]=\{\begin{array}{l}{\Sigma }_{x}E(Y|X=x)p_X(x)\\ {\int }_{\mathbb{R}}E(Y|X=x)f_X(x)dx\end{array}$$